Ray Optics and Optical Instruments formulas

Class 12 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 9, the special cases questions are built on, and diagrams where they help.

55 formulas8 sectionsClass 12 · Chapter 93 of 8 sections free

By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE

Most used formulasOther formulas and cases

1Plane mirror

$$\begin{array}{l}\displaystyle \delta=180^\circ-2i\\[5pt]\displaystyle \text{mirror turned by }\theta\ \Rightarrow\ \text{ray turns by }2\theta\end{array}$$

Image: virtual, same size, as far behind as object is in front, lateral inversion. $\hat e_2=\hat e_1-2(\hat e_1\cdot\hat n)\hat n$ (vector law): only the component along the normal reverses.

$$n=\frac{360^\circ}{\theta}-1\ \ (\text{if }360/\theta\text{ even})$$

Images between two mirrors at angle $\theta$. If $360/\theta=m$ is odd: $m-1$ (object on bisector), $m$ (otherwise). If fractional: integer part. Parallel mirrors: infinite. All images lie on a circle through the object centred at the junction.

Velocity of image: normal component reverses, parallel component same. Object moves towards the mirror at $v$ → image approaches the object at $2v$. Mirror moves at $v$ → image moves at $2v$.
Minimum mirror height to see full self: $h/2$ (any distance). To see a wall behind (person at room centre): $1/3$ of the wall height.

2Spherical mirrors

$$\frac1v+\frac1u=\frac1f,\qquad f=\frac R2,\qquad m=\frac{h_i}{h_o}=-\frac vu=\frac{f}{f-u}=\frac{f-v}{f}$$

$$\begin{array}{l}\displaystyle \frac1v+\frac1u=\frac1f\\[6pt]\displaystyle f=\frac R2\\[6pt]\displaystyle m=\frac{h_i}{h_o}=-\frac vu=\frac{f}{f-u}=\frac{f-v}{f}\end{array}$$

Cartesian signs: distances measured from the pole, positive along incident light. Concave $f<0$, convex $f>0$. $m<0$: inverted, $|m|>1$: enlarged. $f$ does not depend on the medium or on colour.

Object (concave)	Image
∞	at F, real, point-sized
beyond C	between F and C, real, inverted, diminished
at C	at C, real, inverted, same size
between C and F	beyond C, real, inverted, enlarged
at F	at ∞
between F and P	behind mirror, virtual, erect, enlarged

Convex (real object): always virtual, erect, diminished, between P and F. Rear-view mirror (wide field).

$$m_L=-\frac{dv}{du}=-m^2\ \ (\text{mirror}),\qquad m_{\text{area}}=m^2,\qquad x_1x_2=f^2$$

$$\begin{array}{l}\displaystyle m_L=-\frac{dv}{du}=-m^2\ \ (\text{mirror})\\[6pt]\displaystyle m_{\text{area}}=m^2\\[6pt]\displaystyle x_1x_2=f^2\end{array}$$

Longitudinal (short object along axis) and areal magnification; Newton's formula with distances from the focus. Long object: find each end's image separately.

$$v_I=-m^2v_O\ (\text{along axis}),\qquad v_I=m\,v_O\ (\text{perpendicular})$$

$$\begin{array}{l}\displaystyle v_I=-m^2v_O\ (\text{along axis})\\[6pt]\displaystyle v_I=m\,v_O\ (\text{perpendicular})\end{array}$$

Image speed for a moving object (relative to mirror). $P=-\dfrac1f$ (m) for a mirror; concave converging (positive power).

3Refraction at plane surfaces

$$\mu_1\sin i=\mu_2\sin r,\qquad \mu=\frac cv=\frac{\lambda_0}{\lambda},\qquad {}_1\mu_2=\frac{\mu_2}{\mu_1}=\frac1{{}_2\mu_1}$$

$$\begin{array}{l}\displaystyle \mu_1\sin i=\mu_2\sin r\\[6pt]\displaystyle \mu=\frac cv=\frac{\lambda_0}{\lambda}\\[6pt]\displaystyle {}_1\mu_2=\frac{\mu_2}{\mu_1}=\frac1{{}_2\mu_1}\end{array}$$

Frequency unchanged; $v$ and $\lambda$ fall by $\mu$. $\mu\sin\theta$ is constant through parallel layers (only the first and last media matter). Cauchy: $\mu=A+\dfrac{B}{\lambda^2}$ (violet bends most). Reflected ⟂ refracted when $\tan i=\mu$.

$$\text{optical path}=\mu d,\qquad \Delta=(\mu-1)t,\qquad t_{\text{slab}}=\frac{\mu t}{c\cos r}$$

$$\begin{array}{l}\displaystyle \text{optical path}=\mu d\\[6pt]\displaystyle \Delta=(\mu-1)t\\[6pt]\displaystyle t_{\text{slab}}=\frac{\mu t}{c\cos r}\end{array}$$

Waves in thickness $t$: $\mu t/\lambda_0$. Time through a slab at angle $r$ inside.

$$d_{\text{app}}=\frac{d}{\mu},\qquad \text{shift}=d\left(1-\frac1\mu\right),\qquad d_{\text{app}}=\sum\frac{d_i}{\mu_i}$$

$$\begin{array}{l}\displaystyle d_{\text{app}}=\frac{d}{\mu}\\[6pt]\displaystyle \text{shift}=d\left(1-\frac1\mu\right)\\[6pt]\displaystyle d_{\text{app}}=\sum\frac{d_i}{\mu_i}\end{array}$$

Object in denser medium seen from air (near-normal). Object in air seen from inside medium: appears at $\mu d$ (farther). Image speed: $v/\mu$ and $\mu v$ respectively.

$$\text{Normal shift}=t\left(1-\frac1\mu\right),\qquad \text{Lateral shift }x=\frac{t\sin(i-r)}{\cos r}$$

$$\begin{array}{l}\displaystyle \text{Normal shift}=t\left(1-\frac1\mu\right)\\[6pt]\displaystyle \text{Lateral shift }x=\frac{t\sin(i-r)}{\cos r}\end{array}$$

Glass slab: object appears shifted towards the observer (independent of slab position). Emergent ray parallel to incident. Small $i$: $x\approx ti\left(1-\frac1\mu\right)$.

Slab with back silvered: mirror appears at $t/\mu$ behind the front face; image of an object at $x$ is at $x+2t/\mu$ from the front face.
Variable $\mu(y)$: use $\mu\sin\theta=$ constant with $\tan$ of slope; ray bends towards higher $\mu$.

5 more sections and 33 formulas in the full chapter

4Total internal reflection5 formulas · 1 diagram
5Refraction at a spherical surface2 formulas · 1 diagram
6Thin lenses12 formulas · 1 case table · 1 diagram
7Prism and dispersion8 formulas · 1 case table · 2 diagrams
8Eye and optical instruments6 formulas · 1 case table

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